Abstract
For convex parametric optimization problems it is shown that the optimal solution is directionally differentiable provided that a strong second-order sufficient optimality condition and Slater's condition are satisfied for the unperturbed problem. This directional derivative is equal to the optimal solution of a certain quadratic programming problem. For the construction of this quadratic problem, a preliminary choice of a “suitable” KKT-multiplier is necessary, which under additional assumptions may be taken as a vertex of the set of KKT-multipliers of the unperturbed problem. In the last part of this paper, the contingent derivative of the optimal solution is investigated.
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Dempe, S. Directional differentiability of optimal solutions under Slater's condition. Mathematical Programming 59, 49–69 (1993). https://doi.org/10.1007/BF01581237
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DOI: https://doi.org/10.1007/BF01581237